In this work, differential geometry of the Z-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.
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The paper investigates gradings of complex simple Lie algebras, focusing on -gradings and their algebraic structures.
problem Investigating the algebraic structure of -gradings of complex simple Lie algebras.
method Completely determining the possible reductive algebras and proving the uniqueness of a specific free nilpotent Lie algebra.
result The only free nilpotent Lie algebra of step 3 that appears as the negative part of a -grading is the usual -grading of the exceptional Lie algebra .
In this work, the Z-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algeb…
The article explores causal structures in symmetric spaces and their relation to AQFT.
problem Understanding causal structures in symmetric spaces and their applications in AQFT.
method Classification of reductive causal symmetric spaces using Euler elements and 3-grading.
result Extraction of real Matsuki crowns and description of stabilizer groups of Euler elements.
We apply the Rasmussen spectral sequence to prove that the -graded vector space structure of the HOMFLYPT homology over detects unlinks. Our proof relies on a theorem of Batson and Seed stating that the -graded vector space structure of the Khovanov homology over $\mathbb{Z}_2…
In arXiv:1308.3152, the author proved that the Khovanov-Rozansky homology with potential is an invariant for transverse links in the standard contact -sphere. In the current paper, we study the -graded -module structure of $\mathcal…
Study modular geodesics and wedge domains in non-compactly causal symmetric spaces.
problem Understanding the geometric implementation of modular group in symmetric spaces.
method Analyzing the flow generated by Euler elements and their geometric properties.
result The wedge region W is connected and coincides with the observer domain under certain conditions.
We construct the generalized version of covariant Z_3-graded differential calculus introduced by one of us (R.K.), and then extended to the case of arbitrary Z_N grading. Here our main purpose is to establish the recurrence formulae for the N-th power of covariant q-differential D_q = d_q + A and to analyze more closel…
We define a homology for closed braids by applying Khovanov and Rozansky's matrix factorization construction with potential . Up to a grading shift, is the HOMFLYPT homology defined in arXiv:math/0505056. We demonstrate that, for , is a $\mathbb{Z}_2\o…